Admissible Reciprocally Symmetric Costs: Combiner Existence and Classification

We classify the continuous reciprocally symmetric cost functions J:(0,∞)→R with J(1)=0 and strictly convex log-substitution G(t):=J(et) (admissible costs) for which the symmetric compound J(xy)+J(x/y) depends on (x,y) only through (J(x),J(y)). We first prove that this dependence is automatic: for every admissible J, there exists a unique continuous combiner (the auxiliary function P that encodes the compound) P:[0,∞)2→R with J(xy)+J(x/y)=P(J(x),J(y)) for all x,y>0 (Theorem 1); P is symmetric, non-negative, satisfies P(u,0)=2u, and inherits monotonicity and coercivity from admissibility. When P is required to be a polynomial, a growth rate comparison between two recursions for G forces degP≤2 (Theorem 4), so P(u,v)=cuv+2u+2v with c≥0, and the corresponding admissible costs are exhausted by two explicit families (Theorem 8)—the hyperbolic family J(x)=c−1(xλ+x−λ)−2c−1 (c,λ>0) and the degenerate quadratic family J(x)=a(lnx)2 (a>0)—with the latter arising as the Inönü–Wigner contraction λ→0+, λ2/c→a of the former (Theorem 9). Two regularity extensions are obtained: a Lebesgue-measurable cost satisfying explicit regularity hypotheses admits a continuous representative (Theorem 5), and in the entire finite-order regime, the diagonal combiner Q(u):=P(u,u), when polynomial of degree d′, obeys the sharp bound d′≥2ρ (Theorem 6), attained with equality in both classified families. The normalisations P(1,1)=6 and G′′(0)=1 single out the canonical representative Jcost(x)=12(x+x−1)−1.